The Reproducing Kernel Structure Arising from a Combination of Continuous and Discrete Orthogonal Polynomials into Fourier Systems

We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier-type systems.We prove Ismail’s conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh’s theory of linear transformations in Hilbert space. The results are illustrated with Fourier kernels with ultraspherical, their continuous q-extensions and generalizations. As a byproduct of this approach, a new class of sampling theorems is obtained, as well as Neumann-type expansions in Bessel and q-Bessel functions.